The News and Information Publication of The Society of Volume 76 Number 2 July 2007

Rheology Bulletin Inside: John Brady is 2007 Bingham Medalist SOR2007: Salt Lake City, Utah USA Phase Angle in Oscillatory Testing ICR2008 Monterey USA Technical Program 1 How do I know if my phase angles δδ are correct ? δδ? ? Sachin S. Velankar, University of Pittsburgh and David Giles, University of Minnesota

validating phase angle measurements close to 90 degrees. 1. Introduction This is important for practical purposes, not least because δ approaching 90 degrees corresponds to the terminal A large fraction of rheological testing involves small- region of most viscoelastic materials, where significant amplitude oscillatory shear, sometimes referred to as connections between the rheology and structure can be “dynamic” experiments, which yield the complex modulus made. For example, such as entangled polymer G* of the sample. The results of such tests are often ′ melts and polymer solutions generally show “standard” represented in terms of the storage and loss moduli G and ′ ω2 ′′ ω ′′ -like terminal behavior (G ~ and G ~ ), which G , but it is conceptually useful to instead think of G* in can be often be related quantitatively to specific δ terms of its magnitude |G*| and its phase angle . How microscale dynamic processes. In other cases, e.g. block can these quantities be validated? If a reports, copolymers, gels, or particle-filled systems, we are often δ say, |G*| = 12,755 Pa, and = 88.78 degrees, how closely interested in deviations from standard liquid-like terminal can the user trust these values? behavior to help identify, for example, an order-disorder 6 The magnitude |G*| can be validated easily with transition or a liquid-to-gel transition. A recent article in Newtonian calibration standards of known . the Rheology Bulletin has also discussed the challenge of There have also been excellent studies of the measuring terminal viscoelastic properties in the context reproducibility |G*| of molten polymers, which is of calculating the relaxation function G(t). 1-5 important for quality control applications of . When attempting to validate phase angle measurements, a Even in the absence of any rigorous validation, the significant problem is the choice of a viscoelastic standard specifications on the torque, angular displacement, and for calibration: while viscosity standards are readily frequency range of the rheometer provide guidance on available, materials with standard viscoelastic properties when the range or sensitivity of the instrument is are less easy to come by. In the absence of standards, exceeded, and hence large errors in |G*| can be expected. some laboratories validate a δ = 90 degrees using a In contrast, it is much more difficult to judge the accuracy Newtonian , δ = 0 degrees using a steel test specimen, of the phase angle, δ. We are not aware of studies that and presume that phase angles between 0 and 90 degrees evaluate the accuracy of phase angle measurements. are accurate. Some rheology labs have a standard PDMS Furthermore, rheometer manufacturers do not provide putty with a broad spectrum of relaxation times whose specifications for phase angle, and hence there is no clear phase angle at a particular temperature and frequency has guidance on when the range or resolution of the rheometer been specified by the supplier. The National Institute for is exceeded. It is not difficult to understand why phase Standards and Technology (NIST) has developed standard angle specifications are generally missing: phase angle reference materials for viscoelastic measurements, most resolution and accuracy will surely depend on the quality recently the SRM 2490 and SRM 2491 polymer solution 7,8 of the torque and angular displacement signals. and melt . However most laboratories do not have these Approaching the low end of the torque or displacement fluids, and some questions have also been raised about the range, it might still be possible to measure the average validity of the viscoelastic specifications given for SRM 9 magnitudes of these signals (allowing determination of 2490 . It would be very useful for experimental |G*| within some well-defined, reasonable certainty), while rheologists to have a testing protocol, along with different the signal quality becomes too poor to accurately resolve viscoelastic calibration standards, to validate oscillatory the phase shift between them. How can a rheologist measurements on their own . establish conditions under which his or her rheometer can Here we show that a linear, monodisperse, and well- measure phase angles within some well-defined certainty? entangled polymer melt can serve as an excellent A broad validation of phase angle measurements is beyond viscoelastic calibration standard when δ is close to 90 the scope of this short article. We will concentrate on degrees. Such a material has a sharp transition to its

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Rheology Bulletin, 76(2) July 2007  terminal region as frequency is reduced (see details ), and d G* Similarly, dG′′ dδ (7) thus is in the terminal region even when δ is still far from = + ′′ * ()δ 90 degrees, and hence still easy to measure. Phase angles G G tan close to 90 degrees can then be validated by verifying that the rheometer reproduces the expected terminal behavior In Eqs. 6 and 7 above, the left hand side is the fractional (tanδ proportional to 1/ω) at lower frequencies. error in G′ and G″, the first term on the right hand side is the fractional error in the magnitude of the complex In essence, this phase angle validation is a test of self- modulus, and the last term reflects the error in measuring consistency of the rheometer: (1) use the rheometer to phase angle. It is clear that as δ approaches 90 degrees, δ characterize the terminal behavior of the fluid when is the fractional error in G′ grows without bound even when not very close to 90 degrees, and (2) use the now- the error in phase angle dδ remains finite. This is just a characterized terminal behavior to test the rheometer quantitative way of stating that it is difficult to δ performance under adverse conditions (e.g. approaching characterize the of weakly-viscoelastic materials. 90 degrees closely, lower displacements, lower torques, The situation is reversed when δ approaches 0 degrees: etc.) then the error in G″ becomes large, whereas the error on G′ remains finite. In short, when one modulus is much 2. A brief primer on dynamic oscillatory smaller than the other, little of the total signal comes from measurements the response associated with the smaller modulus, and Theory more error is likely in its measurement. Here, we will focus on the case of δ approaching 90 degrees, where the Most dynamic oscillatory tests are performed in controlled elasticity is becoming very weak. strain mode. A sinusoidally-varying strain: The above equations suggest that rather than δ itself, it is γ = γ ()ω 0 sin t (1) better to work in terms of tanδ (or 1/tanδ) since these ω γ directly relate to the accuracy and precision of the dynamic is imposed on the sample. Here is the frequency and 0 moduli. We will reiterate this point at the end of the is the strain amplitude. The in the sample follows: following section.

σ = σ ()ω + δ = * γ ()ω + δ (2) 0 sin t G 0 sin t Practical measurements σ * where 0 is the stress amplitude and |G | is the magnitude Assuming a parallel plate geometry with plates of diameter of the complex modulus. Most commonly, analysis is 2R and gap of h, oscillatory measurements generally performed in terms of the storage and loss moduli: depend on applying a displacement:

σ = γ []G′sin()ωt + G′′cos ()ωt (3) θ = θ ()ω 0 0 sin t (8)

where G′ = G* cos()δ and G′′ = G* sin()δ (4) and measuring a torque: = ()ω + δ (9) Even if the tests are performed in controlled stress mode, T T0 sin t the same equations can be used to calculate the moduli. Here θ0 and T0 are the displacement and torque amplitudes We can take the first step in error analysis as: respectively. For the parallel plate geometry, one can obtain10: dG′ = cos()δ d G* − G* sin()δ dδ (5) Rθ σ γ = 0 ; σ = 2T0 , and hence * 0 2T0h 0 0 G = = * h πR3 γ π 4θ dG′ d G 0 R 0 = − tan()δ dδ (6) ′ * G G (10) Slightly different equations can be derived for other measurement geometries10. Thus, dynamic oscillatory measurements on rotational rheometers depend on  θ δ. Any linear viscoelastic fluid can be represented as a sum of Maxwell measuring three quantities: θ0, T0 and Conceptually, modes. In a linear, monodisperse, well-entangled polymer, higher order these three quantities can be measured by plotting the relaxation modes are much faster and much weaker than the longest displacement and torque signals, as illustrated in Fig. 1. mode. Therefore the terminal region, i.e. the region in which the longest δ mode dominates, ranges from from tanδ → ∞ (δ = 90 degrees) down to The factors limiting the accuracy of tan are clear from about tanδ ≈ 10 (δ ≈ 84 degrees). In contrast, in most other fluids, this diagram: As with any experimentally-determined higher order modes become important when tanδ is still large (δ is still quantities, there are some errors associated with measuring close to 90 degrees). displacement and torque. For example, if θ0 or T0 are

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Rheology Bulletin, 76(2) July 2007 1 1   t t t

0 0

displacement displacement torque torque -1 -1 0246810 0246810 time time

Fig. 1: a. The zero crossing of the sinusoidal displacement and torque signals give the phase lag, from which the phase angle δ can be calculated. b. If the one or both of the signals are noisy, it will cause some uncertainty in the zero crossing (illustrated by the shaded line), and hence error in measuring δ.

small, the displacement and torque signals can show Once again we note that it is tanδ, and not δ, that is the substantial noise, as illustrated in Fig. 1; such noise was more natural quantity for analysis. Actual implementation indeed evident in our experiments (discussed below) at of Eqs. 11 and 12 change from one instrument to another, small displacements or torques. There may also be minor the raw signals are often digitized before any analysis, and drifts in the instrument baseline (e.g. the residual torque in inertial corrections and baseline subtraction may be an an air bearing) over the timescale of a single oscillation. integral part of the analysis. Regardless of cause, these errors in displacement and torque cause errors in the phase angle. 3. Experiments and results

Early rheologists measured θ0, T0 and δ quite literally as The fluid used was Liquid Isoprene Rubber (LIR50), a illustrated in Fig. 1 by plotting the torque and linear monodisperse 1,4-polyisoprene (high cis content) displacement signals output from the rheometer on chart supplied by Kuraray Corp. This polymer was made by paper (see Appendix 2 in Walters11 for a detailed graphical anionic polymerization and had a molecular weight of procedure). Modern rheometers on the other hand use ~45000 kg/mol and polydispersity of less than 1.1. All signal processing techniques, often in the digital domain, tests were performed using the AR2000 rheometer with a to obtain phase angles from the raw signals. One approach 40 mm or 25 mm parallel plate geometry with gaps is to use a cross-correlation method of beating the torque ranging from 0.5 to 2 mm, at a temperature of 25°C against two reference signals, one in phase with the maintained with a Peltier plate. The rheometer was displacement and the other 90 degrees out of phase with mounted on a vibration-isolated platform and leveled. the displacement10: Samples were loaded without air bubbles and excess sample was “trimmed” from the edges when the gap was π 2 N 10% larger than the desired gap. All tests were performed ω ω S = OTsin()ωt dt = T cosδ ; using the default settings for oscillatory testing 1 π 0 N 0 (“Continuous oscillation”, conditioning time and sampling 2πN time both being 3 s or 1 cycle, whichever is longer). ω ω S = OTcos()ωt dt = T sin δ (11) 2 π 0 N 0 3.1. Validation of phase angles : Constant-strain frequency sweep experiments where N is the number of cycles. Thus, the T 0 and the phase angle can be obtained independently: A common oscillatory test sequence involves conducting a single oscillatory measurement at each frequency in a = 2 + 2 S2 specified frequency range, with the strain amplitude being T0 S1 S2 and tan δ = (12) S1 kept constant throughout the frequency range. This test

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Rheology Bulletin, 76(2) July 2007  104 2.1 105 89.99

10% ) 1%  4

10 0.1% *tan fit  strain sweeps 103 log ( log 103 89.9

1.9 89.7 102 0.01 0.1 1 10  (rad/s) 102  89 1 tan

G', G" (Pa) 10 fit region degrees 

100 86 101 fit region 10% 80 1% 10-1 0.1% fit 60 strain sweeps a b 10-2 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100 frequency  (rad/s) frequency  (rad/s)

Fig. 2: Black symbols are a. oscillatory moduli from frequency sweep experiments at three different strain amplitudes. b. tanδ from the same experiment. Red symbols are data from strain sweep experiments of Fig. 3. Thin blue lines are terminal behavior (Eqs. 13) fitted to the 0% strain data within the shaded yellow rectangles. Inset to b. shows the tan δ

results in a form more  suitable to identifying deviation from terminal behavior. In the inset, solid blue line shows the terminal behavior (Eqs. 13), and dotted blue lines show 5% deviation from the terminal behavior.

sequence is often called a “frequency sweep” test in the the correct terminal behavior can be captured from phase software associated with many rheometers. The black angle data far from δ = 90 degrees.  diamonds in Figs. 2 a&b show the results of a frequency sweep test performed at a strain amplitude of 10%, which Upon repeating the frequency sweep tests at lower strains, δ is within the linear viscoelastic limit. As expected, this the rheometer makes significant errors at large tan , e.g. at δ linear, monodisperse, entangled polymer shows terminal 0.1% strain, it is not possible to measure a tan of even δ behavior down to fairly small tanδ values of about 10. 100 ( = 89.4 degrees) accurately. However, small values There is therefore a range of frequencies (illustrated by the of tanδ can still be measured accurately. In other words, yellow region in Figs. 2 a&b) in which two conditions are as tanδ increases (δ approaches 90 degrees), an satisfied: the material is in the terminal regime, and the increasingly larger strain amplitude is necessary for instrument can be expected to measure tanδ reliably. We accurate measurements. This immediately suggests that can therefore establish the terminal behavior of this the accuracy of frequency sweep tests can be improved by  material: increasing the strain amplitude as tanδ increases. This will be discussed in more detail below. 106.7 G′ =13.13ω2 ; G′′ =1401ω ; tanδ = (13) ω

The accuracy of oscillatory measurements as δ approaches 3.2. Instrumental limits for phase angle accuracy: 90 degrees can now be tested by examining deviations A strain-sweep protocol from this behavior at lower frequencies. Indeed the data A key finding of Fig. 2 is that at small strain, there can be do deviate from the expected terminal behavior. Such significant errors in measuring tanδ. It is therefore of deviations are more clearly evident when the product of immediate interest to find the minimum strain, γ , tanδ and frequency – which is expected to be constant in min required for measuring tanδ within a specified error, say the terminal region – is plotted vs. frequency (see inset to 5%. To characterize this γ as a function of tanδ, we Fig. 2b). If we regard a 5% error on tan δ as acceptable, at min conducted strain sweep tests at several different 10% strain it appears that oscillatory measurements are frequencies. Fig. 3 shows a typical sample of the results. accurate up to tanδ on the order of 1000 (δ ≈ 89.94 At each frequency, the measured values of tanδ show a degrees). This simple test to validate phase angle plateau at sufficiently high strain amplitude, and a measurements can be performed only because the fluid is deviation from the plateau at low strain amplitude. We monodisperse and well-entangled, i.e. the phase angle will discuss these features in succession. validation is crucially dependent on the assumption that (continues page 18) 12

Rheology Bulletin, 76(2) July 2007 (Phase Angle, continued from page 12)

10 2.6

% 1 measurements

2.4 min accurate  )  +/- 5% high-strain error plateau 0.1 2.2 log (tan min minimum strain 2 1rad/s 0.01 measurements inaccurate 0.56 rad/s 0.32 rad/s 1.8 0.001 0.01 0.1 1 10 100 1000 0.1 1 10 100 strain  % frequency  (rad/s)

Fig. 3: Typical strain sweep measurements at three Fig. 4: Minimum strain amplitude, γmin, as a function of frequency. Red circles different frequencies. The horizontal lines show +/- are the γmin values obtained from Fig. 3, and the line is a guide to the eye. For ω> 5% error limits for tanδ at 0.56 rad/s. The vertical line 20 rad/s, the γmin was smaller than the smallest strain that could be applied by the γ identifies min, the minimum strain required for rheometer for the geometry used. Data at ω< 0.1 have been excluded because the reliable phase angle measurement. standard deviation on tanδ exceeded 5% (see inset to Fig. 2b).

The mean values of the high-strain plateau at each increase the strain amplitude at low-frequency. The same frequency have been added to Figs. 2a and b (red points). strategy was proposed previously for improving the These mean values agree well with the previous frequency reproducibility of |G*|2. One convenient way of doing so sweep data, and are all within 5% error of the expected is to perform a frequency sweep at fixed stress amplitude, terminal value. Thus, on an average, the AR2000 can so that the strain increases as frequency reduces (γ ~ ω-1 measure large tanδ values – exceeding 1000 – accurately. for a material in its terminal region). Another convenient However, the error bars in Fig. 2b show the standard method is to use a “Minimum Torque” option available in deviation of the various points constituting the plateau. some rheometers for strain-controlled frequency sweep From these error bars it is clear that as tanδ approaches tests. This option increases the strain amplitude beyond and exceeds about 1000 (ω ~ 0.1 rad/s), the standard the specified value if the torque is less than a user- deviation on tanδ increases sharply and can exceed 5% of specified minimum value. It would be useful if the the mean value (this is more clearly visible in the inset to software also had provision to specify a maximum strain Fig. 2b), i.e. a single measurement of tanδ is not likely to that should not be exceeded. In all such cases, when be within 5% error. increasing the strain to improve the quality of oscillatory data, the rheologist must take care to remain in the linear Fig. 3 also shows that with decreasing strain, the measured viscoelastic region. values of tanδ deviate, and reduce systematically from the plateau value. We can immediately identify the strain, The form of Fig. 4, viz. the minimum strain vs. frequency, γ , below which most tanδ values are more than 5% in is not very practical because a different material may show min γ ω error; the procedure for doing so is illustrated at the a significantly different min vs. behavior. For example, frequency of 0.56 rad/s in Fig. 3. The values of γ thus consider what would happen if the same test were repeated min δ obtained are plotted as a function of frequency in Fig. 4. with a less elastic fluid for which tan at 1 rad/s is much This diagram is a map of instrumental limitations when higher, say 10,000. We certainly do not expect the δ measuring phase angles: the region below the solid line, rheometer to be able to measure accurately a tan of labeled “measurements inaccurate” is to be avoided when 10,000 – no matter how high the strain. conducting measurements. Therefore, we redraw Fig. 4 in a form that directly This map quantifies the observation already made in addresses the question: “What conditions should be met to Section 3.1, viz. accurate phase angle measurements measure any particular value of tanδ accurately?” In require increasingly larger strain amplitudes at low drawing such a map, we must avoid “derived” quantities frequencies. Thus, a simple strategy to improving the such as strain, and instead use more basic parameters viz. accuracy of oscillatory measurements is: don’t conduct a the torque and the displacement. Accordingly, a better frequency sweep test at fixed strain amplitude, instead, representation of the instrumental limits may be obtained

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Rheology Bulletin, 76(2) July 2007 1E-2 1E+4 Dataset of Fig. 4 Dataset of Fig. 4

measurements accurate measurements 1E-3 1E+3 additional additional accurate runs runs

1E-4 1E+2 displacement (rad) torque (microN-m) 1E-5 1E+1

1E-6 0.1 1 10 100 1000 1E+0 tan  0.1 1 10 100 1000 tan 

Fig. 5: Maps of instrument limits. Red circles are the torque and displacement corresponding to the same dataset as Fig. 4. Remaining symbols are various runs with different gaps and plate diameters.

by plotting the minimum torque Tmin and minimum In this article we have only been concerned with large displacement θmin corresponding to each γmin, as a function values of tanδ. As mentioned in Section 2, at large tanδ, of the tanδ corresponding to each frequency. Such a oscillatory measurements are limited by the error in G′ representation is shown in Fig. 5, where the red points which grows proportionately to tanδ. In the other extreme correspond to the same dataset in Fig. 4. of tanδ approaching zero, oscillatory measurements are We repeated the analysis of Fig. 3 several times over a limited by the errors in G", which grow proportionately to δ -1 δ θ period of a few months using parallel plate geometry with (tan ) . Thus we anticipate that with decreasing tan , min gaps ranging from 2 mm to 0.5 mm, and plate diameters of and Tmin will rise again; this is the reason why the broad 40 mm or 25 mm. These data are also shown in Fig. 5. yellow curves in Fig. 5 have been shown to have minima. ′ The broad yellow lines through the scattered points show Experiments with other fluids for which G exceeds G" the general trend: measuring higher tanδ requires would be necessary to verify the upturn in the minimum δ increasingly higher amplitudes of the basic torque and torque or displacement for tan < 1. displacement signals for the same (5%) error limits. There Of these two parameters, θmin or Tmin, which one is the is substantial scatter in the minimum torque and more fundamental limiting factor? Our results of Figure 5 displacement, i.e. in some runs, the specified level of show a great deal of scatter, both for small changes in δ performance (5% error on tan ) can be achieved under geometry and even for repeat tests using the same more adverse conditions, whereas in other runs, the same geometry. Therefore we are unable to answer this performance required much larger displacement and question here except to note that the range of scatter is torque values. We are not certain why the instrument larger for the torque than for the displacement. This limits are so inconsistent; it may be due to uncontrollable argues for a displacement limit that is fundamental. factors such as minor changes in air bearing , Repeating these experiments with fluids with substantially incidental vibrations, possible electrical noise, etc. It is different viscosity would help determine whether torque or important to emphasize that it is only the minimum torque displacement is the fundamental limiting factor. and displacement limits of the rheometer that are not δ reproducible from one run to another; the tan values away 4. Closing comments from the limits (i.e. the plateaus in Fig. 3) were highly consistent. For accurate measurements, one must conduct In summary, we have presented procedures to validate the experiments substantially above the minimum limits accuracy of phase angle measurements as the phase angle estimated in Fig. 5 so that irreproducibility of the limits approaches 90 degrees. They are easy to conduct and use does not affect the results. readily available materials, viz. linear, monodisperse, well- entangled polymers, as calibration standards. Therefore

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Rheology Bulletin, 76(2) July 2007 the test can be applied easily by any rheologist to test any provided signals are maintained above limits such as those specific rheometer. of Fig. 5. This is remarkable performance, and better than we had expected prior to performing these tests. This In plotting the instrumental limits in the form of Fig. 5, we level of performance was realized using the default test have presumed that the most important parameters that settings of the rheometer, and without taking any special affect phase angle accuracy are the torque and the precautions in sample loading, except to avoid bubbles: displacement. While this is eminently plausible, other bubbles could give a large error in G' in the terminal factors may also play a role. For example, at high region due to their elasticity. frequency, the accuracy of any inertial corrections, rather than limitations of torque or displacement, may dominate the performance limits. In the experiments in this paper, Acknowledgement inertial effects were negligible over most of the frequency range. We are grateful to Kuraray America Co. for donating the LIR 50 polyisoprene sample for this research. We are also We have used the polyisoprene LIR50, for phase angle grateful to experts from TA Instruments (notably Dr. validation because this is a linear, monodisperse, room Bernard Costello and Dr. Ron Garritano) and Paar-Physica temperature melt which was already available in a large (Dr. Joerg Lauger) for discussions and clarifications, and quantity in our laboratory. Other rheologists may find it to Prof. John Dealy (McGill University) and Dr. Peter more convenient to select other model fluids, either Saucier (Dow Chemicals) for providing references 1-5. polymer melts or solutions, that are more similar to their fluids of interest. As long as the fluid is monodisperse and well-entangled, the terminal region will extend over a Bibliography wide range of tanδ and the procedure outlined here can be applied. In fact, the strain sweep procedure of Fig. 3 does 1. Saucier, P. C. & Obermiller, D. J. Precision in linear not even need a model fluid: it will yield the limits of viscoelastic property measurement: Establishing a protocol resolution of tanδ (Fig. 5) for any material and at any for statistical quality control. Annual Technical Conference phase angle. However, without a model fluid, there will - Society of Plastics Engineers 53rd, 1085-90 (1995). 2. Tchir, W. J. & Saucier, P. C. A statistical approach to curve be no guarantee that the high-strain plateau values of Fig. fitting rheological data inclusive of experimental errors. 3 are correct; whereas with a linear monodisperse fluid Annual Technical Conference - Society of Plastics such as LIR50, the same experiment will also validate the Engineers 49th, 2321-5 (1991). accuracy of tanδ as δ approaches 90 degrees. 3. Saucier, P. C. & Tchir, W. J. Measurement and modeling of viscoelastic properties: estimation of parameters unbiased It is worth noting that in our strain-controlled experiments, by property evolution. Annual Technical Conference - while the AR2000 could measure phase angles quite close Society of Plastics Engineers 50th, 2452-6 (1992). to 90 degrees, it never reported δ values exceeding 90 4. Bafna, S. S. Precision of dynamic oscillatory measurements. degrees (i.e. negative values of tanδ). This is somewhat Polym. Eng. Sci. 36, 90-97 (1996). surprising; the simple-minded expectation would be that 5. Dealy, J. M. & Saucier, P. Rheology in Plastics Quality the rheometer makes random errors in measuring phase Control (Hanser Publishers, Munich, 2000). δ 6. Dealy, J. M. Questions about relaxation spectra submitted angle, and hence as approaches close to 90 degrees, by a reader. Rheology Bulletin 76, 14 (2006). δ measured values of will fall on both sides of 90 degrees. 7. Fanconi, B. & Flynn, K. Non-newtonian polymer solution We are not sure why the measured values of δ remain for rheology now available - Standard Reference Materials. below 90 degrees, but this may also be the reason why Journal of Research of the National Institute of Standards there are systematically-negative deviations of tanδ at low and Technology 108, 97-98 (2003). strain in Fig. 3. 8. Schultheisz, C. R. & Leigh, S. F. in NIST Special Publication 260-143 (U.S. Department of Commerce, The minimum torques and displacements required for National Institute of Standards and Technology, 2002). accurate phase angle measurements (Fig. 5) are 9. Laeuger, J., Heyer, P. & Snyder, C. R. in 77th Annual substantially larger than the minimum torque and Meeting of the Society of Rheology (Vancouver, 2005). displacement specifications quoted by the manufacturer. 10. Macosko, C. W. Rheology: Principles, Measurements, and We suspect that specifications given for the AR2000, and Applications (ed. Macosko, C. W.) (Wiley-VCH, New York, 1994). perhaps for other rheometers, are only intended for G* or 11. Walters, K. Rheometry (Chapman and Hall, London, 1975). η *, and not for phase angle measurements. If so, the given specifications are of only limited use since oscillatory measurements must include the phase angle to give any information about . Despite this criticism, we want to comment that the AR2000 rheometer, and perhaps other comparable instruments, can measure very high values of tanδ – roughly on the order of 1000 – within 5% accuracy,

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Rheology Bulletin, 76(2) July 2007